Generalizable Machine Learning for Mathematical Optimizations

With machine learning gaining increasing popularity in recent years, most successes from employing machine learning methods are found in the fields where the problem solving relies heavily on pattern recognition and representation learning, with solutions consisting of sophisticated rules and logic that cannot be expressed by concise mathematical descriptions. Such fields of study include computer vision, natural language processing, molecular biology, and so on. On the other hand, in the field of mathematical optimization, the common nature of the problems is drastically different: each optimization problem is entirely formulated by highly abstract closed-form mathematical concepts. Therefore, machine learning is seemingly unfit for being a universal tool to mathematical optimizations. While there have been attempts on using machine learning for mathematical optimizations, the majority of the solutions developed in the literature are essentially problem-specific black box models learning existing input-to-solution mappings via brute force data-fitting. Furthermore, many such works only exploit machine learning as computationally faster alternatives or complementary pieces to the conventional mathematical algorithms. In this dissertation, we propose novel and generalizable machine learning approaches, each of which effectively solves a class of mathematical optimization problems, with benefits beyond just having faster or complementary computations. Specifically, we elaborate on three research projects along this direction: the uncertainty injection approach for robust optimizations; the transfer learning with reconstruction loss approach for solving correlated optimization tasks sharing the same input distribution; and the generalization of optimal control and reinforcement learning for non-cumulative objectives, where the proposed generalizations are over both the problem formulations and the corresponding algorithms. Throughout this dissertation, each of the proposed machine learning approaches is designed to be generalizable, flexible, and not restricted to any specific problem or application setting. Each approach serves more than just being a substitution to any existing mathematical optimization algorithm. We hope this dissertation is able to unveil the true potential of machine learning for being an imperative option when it comes to tackling general mathematical optimization problems.

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